2005 paper ijvsmt jpombo jambrosio

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Int. J. Vehicle Systems Modelling and Testing, Vol. 1, Nos. 1/2/3, 2005 79

Copyright 2005 Inderscience Enterprises Ltd.

Dynamic analysis of a railway vehicle in realoperation conditions using a new wheel-railcontact detection model

Joo Pombo

Escola Superior de Tecnologia de Abrantes, Inst. Politcnico de Tomar,rua 17 de 1808 Agosto de, 2200 370 Abrantes, PortugalE-mail: [email protected]

Jorge Ambrsio*IDMEC, Instituto Superior Tcnico,Av. Rovisco Pais, 1049 001, Lisboa, PortugalE-mail: [email protected]*Corresponding author

Abstract: A methodology to create 3D track models, including irregularities, isdeveloped. A new method using wheel and rail surfaces parameterisations isapplied to identify multiple contact points. This method allows lead and lagflange contact, required to study derailment and switches. The formulation canbe used for any kind of wheel or rail profile and still be computationallyefficient, without requiring lookup tables. The method is applied to study theML95 vehicle in a straight track with experimentally measured trackirregularities. The accuracy of this procedure is demonstrated with respect to

selected results obtained by experimental testing and by ADAMS/Rail.

Keywords: railway dynamics; contact point detection; contact models;wheel-rail interaction forces; multibody dynamics.

Reference to this paper to this paper should be made as follows: Pombo, J. andAmbrsio, J. (2005) Dynamic analysis of a railway vehicle in real operationconditions using a new wheel-rail contact detection model, Int. J. VehicleSystems Modelling and Testing, Vol. 1, Nos. 1/2/3, pp.79105.

Biographical notes: Joo Pombo is Invited Professor in Escola Superior deTecnologia de Abrantes, Instituto Politcnico de Tomar and a Post DoctoralResearcher at IDMEC/Intituto Superior Tcnico. He received his Diploma andhis PhD in Mechanical Engineering from the Instituto Superior Tcnico in 1998and 2004 respectively. His areas of interest include multibody dynamics andvehicle dynamics with particular emphasis on railway dynamics.

Jorge Ambrsio is an Associate Professor with Aggregation in InstitutoSuperior Tcnico. He received his PhD in Mechanical Engineering from theUniversity of Arizona in 1991 and his Aggregation degree in 2003 from theInstituto Superior Tcnico. His areas of interest include multibody dynamics,structural dynamics, biomechanics, road vehicle dynamics and railwaydynamics. He is the author of several books and over 300 papers ininternational journals and conferences.


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80 J. Pombo and J. Ambrsio

1 Introduction

Computational tools, besides being required as a design instrument during theresearch and development of new transportation solutions, are also used to studyproblems related to the maintenance and operation of existing railway vehicles. The useof profiledflanged steel wheels running on steel tracks in order to simultaneouslyprovide support, guidance and traction was a brilliant concept in the early days of thisindustry. Nevertheless, the simplicity of the concept masked the complexity of thecontact phenomenon. In fact, the complex contact forces that develop at the wheel-railinterface strongly influence the dynamic behaviour of a rail guided vehicle. Also thecharacteristics of the vehicle suspension, the masses and inertias of the structuralelements, and the geometry and irregularities of the tracks play a dominant role in thisregard. Despite the complexity of the physical phenomena involved the demands for

increasing speeds, better comfort and greater load capacity do not stop increasing. Suchrequirements raise new problems in control of wheel-rail wear and maintenance ofvehicle stability and reliability in different operating conditions.

The study of a complete railway system involves the construction of threeindependent models: the track model; the wheel-rail contact model; the vehicle model.

In this work, a methodology to create detailed three-dimensional track models,including the perturbations resulting from the track irregularities, is implemented(Pombo, 2004; Pombo and Ambrsio, 2003b, 2003c). A pre-processing strategy is usedto obtain the accurate description of two parameterised curves that represent the geometryof the left and right rails space curves.

The wheel-rail contact model involves the determination of the location of the contactpoints on the profiled surfaces of the bodies, the evaluation of the creepages, i.e., thenormalised relative velocities at the point of contact, and the computation of the contactforces. In this work, the geometry of the wheel and rail surfaces is defined in a parametricform. A new formulation for the accurate prediction of the location of the contact pointson the wheel and rail surfaces, even for the most general three dimensional motion of thewheelset, is developed to support the contact model (Pombo and Ambrsio, 2003c, 2004,2005). An elastic force model to calculate the normal contact force in thewheel-rail interface is used to account for the dissipation of energy during contact(Lankarani and Nikravesh, 1990, 1994). The tangential creep forces and momentsin the wheel-rail contact area are evaluated using alternatively, the Kalker lineartheory (1979, 1990), the heuristic non-linear model (Shen et al., 1983) or the Polachformulation (Polach, 1999).

The computational tool described in this work is applied to the study of the ML95trailer vehicle, which is operated by the Lisbon metro company. First, a general

description of the vehicle model is presented, emphasising the mechanical elements thatare relevant for the studies carried out here. Then, the vehicle performance is studied inreal operating conditions. For this purpose, the existing track between the metro stationsofAnjos and Arroios of the Lisbon metro network, with experimentally measured trackirregularities, is considered for the track model assemblage. Conclusions on the benefitsand drawbacks of the computational tool are drawn by comparison of the dynamicanalysis results with those obtained by experimental testing and with the commercialprogram ADAMS/Rail.


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Dynamic analysis of a railway vehicle in real operation conditions 81

2 Track model construction

The strategy adopted for the construction of the track model starts by having the trackcentreline expressed in a parametric form (Pombo and Ambrsio, 2002, 2003a).A moving Frenet frame, which provides the appropriate track referential at every point, isassociated with the track centreline being its axis defined in the intersections of thenormal, osculating and rectifying planes (Mortenson, 1985). The introduction of the cantangle and its variation along the track has been implemented also. The reference planeused in the spatial curve to define the cant angle is the osculating plane, which coincideswith the horizontal plane in a flat curve.

In order to define realistic track models, the track irregularities, which are measuredand provided by railway companies, are considered. During the track model construction,the spatial curves that characterise the geometry of the left and right rails are treated

separately. The adopted strategy starts by defining a set of nodal points that arerepresentative of the left and right space curves. Then, each curve is parameterised as afunction of its arc length, using piecewise cubic interpolation schemes. The availableparameterisation schemes are the cubic splines, the Akima splines and the shapepreserving splines (Pombo and Ambrsio, 2003b, 2003c).

Recognising the fact that the transformation equations are nonlinear and that it is notcomputationally efficient to calculate the curve vectors during the dynamic analysis, apreprocessor to generate all geometric properties of the left and right rail space curves isimplemented. The result is a database for each rail where all quantities relevant for thedynamic analysis are tabulated as functions of the arc length of the rails. At every timestep, the program interpolates linearly both left and right rails databases in order to obtainall the necessary information to analyse the complex interaction between the wheels andrails. A schematic representation of the methodology used in the railway preprocessor ispresented in Figure 1. The interested reader is referred to the work of (Pombo, 2004;Pombo and Ambrsio, 2003b, 2003c).

3 Wheel-rail contact model

Several authors (Johnson, 1985; Kalker, 1990; Kik and Piotrowski, 1996; Polach, 1999;Shen et al., 1983) studied the contact forces between the wheel and the rail during thedynamic motion and, as a result of their investigations, several computer routines are nowavailable for the calculations of the tangential forces at the contact point given the normalforce and the relative velocities between the bodies. Although the literature offerssolutions for the problem of contact mechanics, detailed descriptions of the surfaces in

contact and of the kinematics of the bodies, are required to make use of such solutions.Because the wheel and the rail have profiled surfaces, the prediction of the contact pointlocation online is not simple, especially when the most general three dimensional motionof the wheelset with respect to the rails is considered. Furthermore, the large amount ofparameters that include the shape of the surfaces in contact, relative contact velocities andphysical properties of the materials, unavoidably lead to complex theories. A multibodymethodology that leads to an accurate description of the wheel-rail contact problem andthat is computationally efficient is proposed here.


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Figure 1 Schematic representation of the railway pre-processor

3.1 Definition of the wheel and rail surfaces

The approach followed here is to consider the surfaces of the wheels and rails as sweepsurfaces, obtained by dragging plane curves on spatial curves. As a result, the problem of

describing the surfaces reduces to the problem of defining plane curves, which representthe cross sections of the wheel and rail. Let four independent surface parameters sr, ur,swand uw be used to define the geometry of the wheel and rail surfaces, as shown inFigure 2. The parametersr represents the arc length of the rail space curve, i.e., it definesthe rail cross section on which the contact point lies, while ur defines the lateral positionof the contact point in the rail profile coordinate system (r,r,r). The parameterswrepresents the rotation of the wheel profile coordinate system (w,w,w) with respect tothe wheelset coordinate system (ws,ws,ws), i.e., it defines the rotation of the contactpoint, while uw defines the lateral position of the contact point in the wheel profilecoordinate system (Shabana et al., 2002a, 2002b). In what follows, the subscripts (.)r and


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(.)w refer to the rail and wheel, respectively, whereas the subscript (.)ws refer to the

wheelset.

Figure 2 Wheel and rail surface parameters

The rail surface is generated by the two dimensional curve that defines the rail profile,when it is moved along the rail space curve. The location of the origin and the orientationof the rail profile coordinate system, defined respectively by the vector rr and thetransformation matrix Ar, are uniquely determined, using the surface parametersr(Berzeri et al., 2000). The global position vector of an arbitrary point Q on the railsurface is

,Q Qr r r r = +r r A s (1)

where Qrs is the local position vector that defines the location of the contact point Q on

the rail profile coordinate system, written as:

{0 ( )} ,Q Tr r r r u f u =s (2)

wherefr is the function that defines the rail profile at the cross section. Generally fr is notgiven by simple analytical functions. It is proposed here to parameterise the rail profile asa function of the surface parameter ur using a piecewise cubic interpolation scheme(De Boor, 1978). Hence, the user only has to define a set of control points that arerepresentative of the rail profile geometry. This methodology is general, in the sense thatit allows changing the rail profile if needed. It also allows performing the dynamicanalyses of railway vehicles using rail profiles obtained by measurements or by designrequirements.

The wheel surface of revolution is obtained by a complete rotation, about the wheelaxis, of the two dimensional curve that defines the wheel profile (Shabana et al., 2001).The location of the origin and the orientation of the wheelset reference frame are defined,respectively, by the vector rws and the transformation matrix Aws. The global positionvector of an arbitrary point Q on the wheel surface is written as:

( ),Q Qw ws ws w w w= + +r r A h A s (3)


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where { }1

0 02

T

w H=

h is the local position vector that defines the location of wheelprofile coordinate systems with respect to the wheelset reference frame and H is thelateral distance between wheels profiles origin, Aw is the transformation matrix thatdescribes the orientation of the wheel coordinate system with respect to the wheelsetreference frame, given by (Mortenson, 1985):

cos 0 sin

0 1 0 .

sin 0 cos

w w

w

w w

s s

s s

=

A (4)

The quantity Qws is the local position vector that defines the location of the contact point

Q on the wheel profile coordinate system, written as:

{0 ( )} ,Q Tw w w wu f u =s (5)

wherefw is the function that defines the wheel profile. Since in general, fw is not given byanalytical functions, it is proposed to parameterise the wheel profile using a piecewisecubic interpolation scheme, as described for the rail surface. Hence, the user has to definea set of control points that are representative of the wheel profile geometry. Thismethodology is general and allows using wheel profiles obtained by measurements or bydesign requirements.

The wheel-rail contact model requires a formulation to determine the location of thecontact points. The formulation proposed here demands that the parametric surfaces areconvex. Therefore, when parameterising the wheel profile, it is necessary to avoid thegeometric description of the small concave region in the transition between the wheel

tread and flange. In order to avoid this difficulty, the wheel profile is represented by twoindependent functions twf and

f

wf that parameterise the wheel tread and flange,

respectively. Therefore, the concave region is neglected and the wheel surface is made oftwo convex regions, as shown in Figure 3.

Figure 3 Concave region in the wheel profile

It should be noted that it is not always that the concave region of the wheel profile can beneglected. In the presence of large angles of attack and curves with small radius, ageometrical description of this zone is important. Therefore, if such scenarios areimportant for the application of the methodology, some modifications in the formulationmust be devised to handle concave surfaces also.


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3.2 Algorithm for the wheel-rail contact problem

The accurate prediction of the wheel-rail contact points has to be solved online, for everytime step, during the dynamic analysis. Once the coordinates of the contact points aredetermined, the normal contact forces that develop in the wheel-rail interface have to becalculated. Then, the creepages or normalised relative velocities between the bodies at thepoints of contact, have to be accurately determined. Once the creepages are obtained,several computer routines, using different approaches, can be applied to evaluate thecreep forces. In this algorithm, three distinct methodologies are implemented to calculatethe tangential contact forces that develop at the wheel-rail interface.

A schematic representation of the wheel-rail contact algorithm is presented inFigure 4. The application of this methodology demands the use of an appropriateformulation for the dynamic analysis of multibody systems. The explanation of

such formulation, which is out of the scope of this text, can be found in references(Nikravesh, 1988; Roberson and Schwertassek, 1988). The remaining steps of theprocedure are described hereafter.

Figure 4 Algorithm used to study the wheel-rail contact problem

3.3 Wheel-rail contact points detection

The coordinates of the contact points can be predicted online, during the dynamicanalysis, by determining the four parameters that define the geometry of the contactsurfaces. A two step methodology is used here to determine the coordinates of the contactpoints between wheel and rail surfaces. First, four geometric equations are defined andsolved in order to find the surface parameters that define the coordinates of the candidates


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to be contact points between the surfaces. With reference to Figure 5, these equations are

written as (Pombo and Ambrsio, 2003c, 2004, 2005):1 1

2 2

0 0;

0 0.

T T

r w wr w

T T

r w wr w

t

t

= =

= =

n t d

n t d(6)

where nr is the vector normal to the rail surface,1 2andw wt t are two vectors tangent to the

wheel surface and dwr= rw rr is the distance between the potential points of contact.

Figure 5 Candidates to contact points between two parametric surfaces

The second step of the method consists in the evaluation of the penetration condition inorder to check if the points are, in fact, in contact or not. The penetration conditionspecifies that:

0.Twr r d n (7)

The geometric conditions in equation (6) are four nonlinear equations with fourunknowns, which are the surface parameters sr, ur,sw and uw. This system of equations issolved for every pair of points that are, or may come, in contact. If a pair of points is incontact, then equation (7) is satisfied. Otherwise, vector dwr in equation (7) represents

simply the shortest distance between the two surfaces. In the computationalimplementation, the information of the previous time step is used as an initial estimate forthe solution search of the equations.

For wheel profiles that are worn, or even for some new wheel profiles, there arenonneglectable concave regions in which the points of contact have to be found.Although not presented here, a procedure similar to that described for convex regions canbe devised. However, it should be noted that, not only a new strategy to find the contactpoints between convex and concave regions has to be implemented, but also theevaluation of the normal force that develops has to follow a different contact theory,nonHertzian.


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3.4 The two points of contact scenario

The proposed contact model allows studying two points of simultaneous contact betweenone wheel and the rail by using an optimised search for possible contact points on thewheel tread and on the wheel flange. The methodology takes advantage of the fact thatthe wheel profile is parameterised by two functions twf and

f

wf , as shown in Figure 3.

Since the strategy used to look for the contact points is fully independent for the wheeltread and for the wheel flange, the contact point in the flange does not have to be locatedin the same plane as the contact point in the wheel tread, as shown in Figure 6. Thisfeature allows for the analysis of potential derailments or for the study of the dynamicbehaviour in the presence of switches, among others.

Figure 6 Two points of contact in different planes. Lead contact in the right wheel

3.5 Normal contact forces

The Hertz contact force model with hysteresis damping (Lankarani and Nikravesh, 1990,1994) is used here to calculate the normal contact forces that develop at the wheel-railinterface. This force model accounts for the energy dissipation effect that occurs duringcontact. The normal contact force includes the Hertzian component, which is a functionof the indentation, and a hysteresis damping force component, proportional to thevelocity of indentation, given by:

2

( )

3(1 )1

4neN K

= +

(8)

where is the indentation, n = 1.5 is the parameter used for metal to metal contact,Kisthe Hertzian constant that depends on the surface curvatures and the elastic properties ofcontacting bodies, e is the coefficient of restitution, is the velocity of indentation and

( ) is the velocity of indentation at the initial instant of contact. The velocity ofindentation is evaluated as the projection of the relative velocity vector of the bodies atthe point of contact on the vector normal to the contact surfaces.

Note that equation (8) is only valid for nonconformal contact, where the contact patchis described by an ellipse. For the case of conformal contact, as for instance for wornwheel profiles, either another contact force model must be used or provisions have to bemade to adjust the stiffness K to such conditions. Therefore, in what follows, only


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Hertzian contact is considered. A very promising force model for conformal surfaces, in

wheel-rail contact, is proposed by (Kovalev et al., 2003).

3.6 Tangential contact forces

Knowing the creepages and the normal contact forces, it is possible to compute the creepforces using one of the computer routines available in the literature. In this wheel-railcontact algorithm, three distinct methodologies are implemented as alternatives to eachother. They are the Kalker linear theory, the heuristic non-linear model and the Polachformulation.

According to the Kalker linear theory (Kalker, 1979, 1990) the longitudinal F andlateralF components of the creep force and the spin creep moment M that develop inthe wheel-rail contact region are expressed as:

11

22 23

23 33

0 0

G ab 0

0

cF

F c ab c

M ab c ab c

=

(9)

where G is the combined shear modulus of rigidity of wheel and rail materials and a andb are the semi-axes of the contact ellipse that depend on the material properties and onthe normal contact force. The parameters cij are the Kalker creepage and spin coefficients,obtained in references (Kalker, 1979, 1990) and the quantities , and represent thelongitudinal, lateral and spin creepages at the contact point, respectively. For sufficientlysmall values of creep and spin, the linear theory of Kalker is adequate to determine thecreep forces. For larger values, this formulation is no more appropriate since it does not

include the saturation effect of the friction forces, i.e., it does not assure that .F N

The heuristic non-linear model (Shen et al., 1983) involves calculating the creep force

expected from the Kalker linear theory and modifying it by a factor that takes intoaccount the limiting creep force. First, the resultant creep force of Kalker linear theory iscalculated as:

2 2F F F = + (10)

where the notation (.) now means that the quantities are obtained with the Kalker lineartheory. The limiting resultant creep force is determined by:

2 31 1

; 33 27

; 3

F F FN F N

F N N N

N F N

+

=

(11)

where is the friction coefficient. The new resultant creep force F is used to calculatethe tangential forces as:

; .F F

F F F FF F

= =

(12)


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In the heuristic method the spin creep moment M is neglected. This theory gives more

realistic values for creep forces outside the linear range than the Kalker linear theory.For high values of spin, the heuristic theory can lead to unsatisfactory results(Andersson et al., 1998).

According to the Polach method (Polach, 1999), the longitudinal andlateral components of the creep force that develop in the wheel-rail contact region areexpressed as:

; ,SC C C

F F F F F

= = + (13)

whereFis the tangential contact force caused by longitudinal and lateral creepages, C isthe modified translational creepage, which accounts the effect of spin creepage, and FSis the lateral tangential force caused by spin creepage. The Polach algorithm requires asinput, the creepages , and , the normal contact force, the semiaxes of the contactellipse, the combined modulus of rigidity of wheel and rail materials, the frictioncoefficient and the Kalker creepage and spin coefficients cij. The Polach algorithm allowsthe calculation of full non-linear creep forces and takes spin into account.

4 Vehicle model

The ML95 trainset, shown in Figure 7, is used by the Lisbon metro company in theiroperations. It is an electrical three car unit composed of two powered end vehicles andone intermediate vehicle that is not powered. A schematic representation of a railwayvehicle multibody model is shown in Figure 8. According to this description, the carbody

rests directly on the secondary suspension and its weight is transmitted downwards to thetrack through the bogie frames, primary suspension elements, axleboxes, wheelsets andwheel-rail interface.

Figure 7 The ML95 trainset


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Figure 8 Schematic representation of a railway vehicle multibody model

Forces arising from the wheel-rail interaction are transmitted upwards through the samecomponents. Due to their high structural stiffness, the carbody, the bogies frames, thewheelsets and the axleboxes are considered here as rigid bodies. These bodies areconnected by a set of kinematic joints, responsible for controlling their relative motion,and by a group of rigid or flexible links that compose the suspensions.

4.1 Rigid bodies

The three dimensional model of the ML95 trailer vehicle, represented in Figure 9, isdescribed using 11 rigid bodies, identified with the numbers from 1 to 11. The initialposition of each body is given by the location of its centre of mass with respect to theglobal reference frame (x,y,z). The mass, the inertia properties with respect to the threeprincipal axes and the initial position of each rigid body are presented in Table 1.The numbers in the first column identify the rigid bodies in the vehicle model shown inFigure 9.

Figure 9 Three dimensional model of the ML95 trailer vehicle


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Table 1 Mass, inertia properties and initial positions of each rigid body

Inertia properties (kg.m2) Initial position (m)

ID Rigid bodiesMass(kg) I/I/I x0/y0/z0

1 Rear wheelset rear bogie 933 461.4/61.6/461.4 1.05/0.0/0.4302 Front wheelset rear bogie 933 461.4/61.6/461.4 1.05/0.0/0.4303 Rear axleboxes rear bogie 176 144.5/2.2/144.5 1.05/0.0/0.4304 Front axleboxes rear bogie 176 144.5/2.2/144.5 1.05/0.0/0.4305 Rear bogie frame 1,982 1398.5/2667.0/2667.0 0.00/0.0/0.4606 Rear wheelset front bogie 933 461.4/61.6/461.4 10.05/0.0/0.4307 Front wheelset front bogie 933 461.4/61.6/461.4 12.15/0.0/0.4308 Rear axleboxes front bogie 176 144.5/2.2/144.5 10.05/0.0/0.4309 Front axleboxes front bogie 176 144.5/2.2/144.5 12.15/0.0/0.43010 Front bogie frame 1,982 1398.5/2667.0/2667.0 11.10/0.0/0.46011 Carbody 11,160 14953/225365/224995 5.55/0.0/1.849

4.2 Kinematic joints

The kinematic joints connect the bodies of the multibody system in order to restrain someof their relative motions (Nikravesh, 1988). In the multibody model of the ML95 trailervehicle, four revolute joints are used to connect each pair of axleboxes to a wheelset, onlyallowing the relative rotation between the connected bodies, representing in this form theroller bearings that exist at the extremities of each wheelset. All other relative motionsbetween the system components are limited by force elements.

4.3 Primary suspension model

The primary suspension of a railway vehicle consists of a group of flexible links thatensure the transmission of forces between the bogie frame and the axleboxes assembledat the extremities of the wheelsets. In the ML95 trailer vehicle, the primary suspension ismodelled with a three dimensional spring-damper element, which has stiffness anddamping properties defined for the vertical, longitudinal and lateral directions shown inFigure 10. The characteristics of these spring and damper elements, the numbers of thebodies that they connect and the local coordinates of the attachment points are presentedin Table 2. This table only presents the characteristics of the suspension elements for therear bogie. The front bogie has exactly the same characteristics.

Figure 10 Primary suspension model of the ML95 trailer vehicle


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Table 2 Primary suspension elements used in the model of the ML95 trailer vehicle

Attachment pts local coords (m)

Bodies Body i Body jSuspensionelementdirection

Springstiffness106N/m

Dampingcoef.

103N.s/m Bogie location i j i/i/i j/j/j

Rear right 3 5 0.0/0.9/0.0 1.05/0.9/0.22

Rear left 3 5 0.0/0.9/0.0 1.05/0.9/0.22

Front right 4 5 0.0/0.9/0.0 1.05/0.9/0.22

Vertical K1 1.28 C1 55.41

Front left 4 5 0.0/0.9/0.0 1.05/0.9/0.22

Rear right 3 5 0.0/0.9/0.0 0.775/0.9/0.03

Rear left 3 5 0.0/0.9/0.0 0.775/0.9/0.03

Front right 4 5 0.0/0.9/0.0 0.775/0.9/0.03

Longitud. K2 6.21 C2 86.29

Front left 4 5 0.0/0.9/0.0 0.775/0.9/0.03Rear right 3 5 0.0/0.9/0.0 1.325/0.9/0.03

Rear left 3 5 0.0/0.9/0.0 1.325/0.9/0.03

Front right 4 5 0.0/0.9/0.0 1.325/0.9/0.03

Longitud. K3 6.21 C3 86.29

Front left 4 5 0.0/0.9/0.0 1.325/0.9/0.03

Rear right 3 5 0.0/0.9/0.0 1.05/1.1/0.03

Rear left 3 5 0.0/0.9/0.0 1.05/1.1/0.03

Front right 4 5 0.0/0.9/0.0 1.05/1.1/0.03

Lateral K42.06 106

C470.29 103

Front left 4 5 0.0/0.9/0.0 1.05/1.1/0.03

4.4 Secondary suspension model

The secondary suspension of a railway vehicle consists of a set of springs and dampersthat transmit the forces between the carbody and the bogies providing, in the process, theisolation of the passenger compartment from the vibrations induced by the track. In theML95 vehicle model, the carbody is supported by the two bogies through four airsprings.These components are modelled here with three pairs of spring-damper elements actingin the vertical, longitudinal and lateral directions, as shown in Figure 11. In parallel withthe airsprings, four vertical dampers are used for the stabilisation of the carbody. Thecharacteristics of the spring and damper elements that compose the secondary suspensionmodel, the numbering of the bodies that they connect and the local coordinates of theattachment points are presented in Table 3.

Figure 11 Secondary suspension model of the ML95 trailer vehicle


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Table 3 Secondary suspension elements used in the model of the ML95 trailer vehicle

Attachment points local coords (m)Bodiesconnected Body i Body jSuspension

element


Dampingcoeffic.

103N.s/m i j i/i/i j/j/j

5 11 0.0/0.935/0.09 5.55/0.935/1.0095 11 0.0/0.935/0.09 5.55/0.935/1.009

10 11 0.0/0.935/0.09 5.55/0.935/1.009

Airspringvertical

K5 250 C5 47.54

10 11 0.0/0.935/0.09 5.55/0.935/1.0095 11 0.2/0.935/0.38 5.55/0.935/1.0095 11 0.2/ 0.935/0.38 5.55/0.935/1.009

10 11 0.2/0.935/0.38 5.55/0.935/1.009

Airspringlongitudinal

K6 75 C6 26.04

10 11 0.2/0.935/0.38 5.55/0.935/1.0095 11 0.0/1.135/0.38 5.55/0.935/1.0095 11 0.0/1.135/0.38 5.55/0.935/1.009

10 11 0.0/1.135/0.38 5.55/0.935/1.009

Airspringlateral

K7 75 C7 26.04

10 11 0.0/1.135/0.38 5.55/0.935/1.0095 11 0.48/1.130/0.07 5.07/1.130/1.0095 11 0.48/1.130/0.07 6.03/1.130/1.009

10 11 0.48/1.130/0.07 6.03/1.130/1.009

Verticaldamper

C8 21.00

10 11 0.48/ 1.130/0.07 5.07/1.130/1.009

4.5 Bogie-carbody connection model

In railway vehicles, the bogie-carbody connection is enforced by a group of flexible linksthat have steering functions and that do not transmit vertical loads. In the ML95 trailer

vehicle, the transmission of traction and braking efforts between each bogie and thecarbody is done by two traction rods. These traction rods are modelled using springelements, as depicted in Figure 12. Each spring connects the bogie frame to the centreplate acting in the longitudinal direction. The lateral stability of the carbody is obtainedby the action of two transversal dampers assembled between the centre plate and eachbogie frame. The characteristics of the spring and damper elements, which compose thebogie-carbody connection model, the numbering of the bodies connected by them and thelocal coordinates of the attachment points are presented in Table 4.

Table 4 Bogie-carbody connection elements used in the model of the ML95 trailer vehicle

Attachment points local coords (m)Bodiesconnected Body i Body jConnection

element


Dampingcoeffic.

103N.s/m i j i/i/i j/j/j

5 11 0.245/0.44/0.21 5.795/0.44/1.5995 11 0.245/0.44/0.21 5.305/0.44/1.59910 11 0.245/0.44/0.21 5.305/0.44/1.599

Traction rod K9 1.905

10 11 0.245/0.44/0.21 5.795/0.44/1.5995 11 0.225/0.285/0.27 5.325/0.04/1.5995 11 0.225/0.285/0.27 5.775/0.04/1.59910 11 0.225/0.285/0.27 5.775/0.04/1.599

Transversaldamper

C10 12.0

10 11 0.225/0.285/0.27 5.325/0.04/1.599


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Figure 12 Bogie-carbody connection model for the ML95 trailer vehicle

4.6 Wheel and rail profiles

The wheel profiles for the ML95 trainset, with the geometric characteristics shown inFigure 13, are used to exemplify the representation of the geometric features of a generalwheel. The tracks where the ML95 vehicle travels are composed of UIC 50 rails. Theprofile of these rails, shown in Figure 14, is also used to describe the geometric featuresof rails.

Figure 13 Wheel profile of the ML95 trainset. By courtesy of: Lisbon subway company


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Figure 14 Rail profile UIC 50. By courtesy of:Lisbon subway company

Since the wheel and rail profiles are not given by analytical functions, they areparameterised as functions of the respective surface parameters uw and ur, using cubic,Akima or shape preserving splines. In Table 5 the coordinates of the 11 nodal points thatare used to parameterise the wheel tread profile are presented. The same type ofinformation is presented in Table 6 for the 21 nodal points that describe the wheel flangeprofile. The coordinates in both tables are local since they are defined with respect to thewheel profile coordinate system (w,w), which is depicted in Figure 2. The wheel profile,parameterised using three piecewise cubic interpolation schemes, is shown in Figure 15and it is used hereafter to generate the wheels surfaces that are used in the dynamicanalyses performed.

Table 5 Nodal points used to parameterise the wheel tread profile of the ML95 trainset

Pt w(m) w(m)

1 0.034 0.4317

2 0.025 0.4313

3 0.015 0.4308

4 0.005 0.4303

5 0.005 0.4298

6 0.015 0.4293

7 0.025 0.4288

8 0.035 0.4283

9 0.045 0.427810 0.550 0.4273

11 0.065 0.4268


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Table 6 Nodal points used to parameterise the wheel flange profile of the ML95 trainset

Pt w(m) w(m)

1 0.0690 0.4458

2 0.0680 0.4484

3 0.0670 0.4502

4 0.0660 0.4517

5 0.0650 0.4529

6 0.0640 0.4540

7 0.6276 0.4551

8 0.0610 0.4564

9 0.0595 0.4571

10 0.0580 0.457611 0.0565 0.4579

12 0.0550 0.4580

13 0.0535 0.4579

14 0.0520 0.4576

15 0.0497 0.4567

16 0.0470 0.4552

17 0.0450 0.4535

18 0.0430 0.4514

19 0.0410 0.4484

20 0.0398 0.4457

21 0.0340 0.4298

Figure 15 Wheel profile of the ML95 trainset parameterised with cubic, Akima and shapepreserving splines


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The coordinate points of the wheel profile, provided in Tables 5 and 6, correspond to a

new wheel. Moreover, the approximations of the wheel profile given by all interpolationschemes considered provide similar geometries, as observed in Figure 15. In Table 7 thecoordinates of the 29 nodal points that are used to parameterise a new rail profile UIC 50are presented. These coordinates are defined with respect to the rail profile coordinatesystem (r,r), represented in Figure 2. The nodal points are parameterised using any ofthe piecewise cubic interpolation schemes available to obtain the function fr(ur). Theparameterised rail profile, shown in Figure 16, is used hereafter for the applicationsstudied here. Note that the approximation of the rail profile geometry is similarlyobtained with any of the different interpolation schemes suggested.

Table 7 Nodal points used to parameterise the rail profile UIC 50

Pt r(m) r(m)

1 0.03236 0.03688

2 0.03195 0.03874

3 0.03126 0.04052

4 0.03033 0.04218

5 0.02916 0.04368

6 0.02779 0.04500

7 0.02623 0.04610

8 0.02454 0.04697

9 0.02273 0.04757

10 0.02086 0.04791

11 0.01670 0.0483012 0.01253 0.04861

13 0.008355 0.04883

14 0.004178 0.04896

15 0.000 0.04900

16 0.004178 0.04896

17 0.008355 0.04883

18 0.01253 0.04861

19 0.01670 0.04830

20 0.02086 0.04791

21 0.02273 0.04757

22 0.02454 0.04697

23 0.02623 0.04610

24 0.02779 0.04500

25 0.02916 0.04368

26 0.03033 0.04218

27 0.03126 0.04052

28 0.03195 0.03874

29 0.03236 0.03688


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Figure 16 Rail profile UIC 50 parameterised with cubic, Akima and shape preserving splines

5 Dynamic analysis of the railway vehicle in real operating conditions

All methodologies proposed here are implemented in the general purpose multibody codeDAP-3D (Nikravesh, 1988) that is used for the dynamic analysis of railway and othertypes of rail guided vehicles.

5.1 Simulation scenario

The ML95 trailer vehicle is studied in real operating conditions when travelling betweenthe Lisbon metro stations ofAnjos and Arroios. The track between such stations has astraight geometry with a length of 666 m. The track irregularities are obtainedexperimentally with a vehicle of the Lisbon metro company equipped for the purpose.The data collected consists of a table where all irregularities parameters are stored incolumns as functions of the travelled distance along the track. The information about thetrack geometry and its irregularities is used as input for the railway preprocessor in orderto build a realistic track model. During the track model assemblage, a distance betweencontrol points of 1 m is used and the distance step adopted for the left and right railsdatabases construction is 0.05 m. The rails space curves are parameterised with shapepreserving splines (Pombo, 2004).

When the vehicle model is assembled on the track, the system should be in staticequilibrium in order to avoid discontinuities in the results when the dynamic analysisstarts. With this purpose, the vehicle is assembled on a perfect track withoutirregularities. For this reason, no track irregularities are considered in the first 50 m of thetrack model.


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5.2 Proposed model vs. the commercial software ADAMS/Rail

In this section, the dynamic analysis results, obtained with the methodology proposedhere and implemented in the multibody program DAP-3D, are compared with the resultsobtained with the commercial program ADAMS/Rail (MDI, 1995, 1999) for the samescenario. The strategy adopted for the construction of the ML95 vehicle model inADAMS/Rail follows the same multibody approach described here. The only differenceis related to the fact that in ADAMS/Rail airspring models are available, whereas inDAP-3D these elements are modelled with the system of springs and dampers depicted inFigure 11.

As in DAP-3D, the track model in ADAMS/Rail has a straight geometry and themeasured track irregularities are considered. The wheel-rail contact model used inADAMS/Rail simulation is the Table Book method (MDI, 1995, 1999). In DAP-3D the

creep forces are computed by the Polach formulation. Under such conditions, a dynamicanalysis is carried out with the two computational tools.Figure 17 represents the lateral contact forces on the right wheel of the leading

wheelset of the ML95 vehicle. There is a reasonable agreement between the results,which present identical medium values and similar variation ranges, the ADAMS/Railcontact forces being slightly higher. Also the peaks in the results, arising from the trackirregularities, are larger in the commercial code. The differences between the results canbe explained by two main facts. The first one is related with the fact that the creep forcemodels used are not the same. The second reason is due to the different approaches usedby the two programs when including the irregularities in the track model. This suggeststhat the isolated higher peaks in the forces are a result of the track parameterisationprocess rather than being a physical characteristic of the system.

Figure 17 Lateral contact forces on the right wheel of the leading wheelset


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The normal contact forces on the right wheel of the vehicle leading wheelset are

presented in Figure 18. It is observed that the results present identical medium valueswhen using both codes but the range variation of the normal forces computed by DAP-3Dis higher than the one obtained with the commercial code. Besides the two main reasonspreviously described, another important factor that contributes to the difference is relatedto the different formulations used to calculate the normal contact forces that develop atthe wheel-rail interface. Notice that the results are almost coincident at the beginning ofthe analysis, where no track irregularities are considered. The vertical accelerations onthe carbody of the ML95 vehicle are presented in Figure 19 where a good agreementbetween the results is observed.

Figure 18 Normal contact forces on the right wheel of the leading wheelset

Figure 19 Vertical accelerations on the carbody of the ML95 trailer vehicle


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5.3 Proposed model vs. experimental data

To validate the models and methodologies developed, the results obtained with DAP-3Dprogram are compared with the experimental results in terms of the carbodyaccelerations. The lateral and vertical accelerations on the carbody of the ML95 vehicleare presented in Figures 20 and 21, respectively. The graphics show a reasonableagreement between the results, which present identical medium values and similarvariation ranges. The exceptions are the kinks in the results arising from the trackirregularities. It is observed that, especially in the vertical accelerations, suchdiscontinuities are significantly higher in the results obtained experimentally.

Figure 20 Lateral accelerations on the carbody of the ML95 trailer vehicle

Figure 21 Vertical accelerations on the carbody of the ML95 trailer vehicle


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In order to study the frequency contents of the results presented in Figures 20 and 21, a

Fast Fourier Transform (FFT) is applied to these signals. The spectra for the lateral andvertical accelerations obtained experimentally and with DAP-3D, are presented inFigures 22 and 23, respectively. The lateral acceleration spectra show a good agreement.Moreover, the most meaningful frequency of the experimental results is 0.9 Hz whereasfor the numerical simulation results it corresponds to 1.1 Hz. For the verticalaccelerations the spectra exhibit a reasonable agreement.

Figure 22 Spectra of the lateral accelerations on the carbody of the ML95 trailer vehicle

Figure 23 Spectra of the vertical accelerations on the carbody of the ML95 trailer vehicle


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The most meaningful frequencies for the experimental and numerical results are about

1.2 Hz and 1.7 Hz, respectively. Notice that the most meaningful frequencies exhibithigher values in the results obtained with DAP-3D. This suggests that the suspensions ofthe vehicle model described here are stiffer than the real ones. However, other sources ofdiscrepancies cannot be discarded when comparing experimental with numericalresponses. For instance, errors related with the methods used to measure trackirregularities or limitations associated with experimental procedures can be responsiblefor some of the inconsistencies. The assumptions that the rail profile does not changealong the track or that the vehicle wheels are new are also responsible for some of thedifferences. Other sources of errors include the flexibility of the carbody and the smallvariation on the forward velocity of the vehicle during the data acquisition process. Allthe eventual sources of errors are mostly unrelated to the methods or modelsimplemented but they reflect the limitations of both models and data acquisition

procedures.

6 Conclusions

The study of a railway system involves the construction of three independent models: themodel of the vehicle the model of the track the wheel-rail contact model.

In the first stage, these models are built separately and, afterwards, they areassembled in order to obtain the complete railway system. The vehicle model was buildhere with a multibody formulation using only rigid bodies since it was assumed that theflexibility of the structural elements did not play a role in the dynamic behaviour of thevehicle. The bodies were connected by a set of flexible links that were used to model theprimary and secondary suspensions. The mechanical properties of such elements weredescribed in detail.

The methodology used for the track model construction allows for the accuratedescription of fully three dimensional track geometries. It also allows simulating realistictracks since the measured irregularities are considered by the formulation during the trackmodel assemblage. The methodology developed is appropriate to the needs of the railwayoperators since the required input data is the information that is standard in the railwayindustry. Another advantage of the proposed track model is that it makes the timerequired for the dynamic simulation of the rail-guided vehicles completely independentof the track complexity and of the type of parametric interpolating curve used. Anydescriptive form of parametric curves is dealt within the preprocessor while the dynamicanalysis program only has to proceed with linear interpolations of the rail databases. Byensuring that the arc length step is small enough, the linear interpolation procedure does

not introduce any significant error in the geometric description of the rails.A general formulation for the accurate prediction of the location of contact points on

the wheel and rail surfaces was proposed and implemented. With this approach, thecoordinates of the contact points can be predicted online during the dynamic simulationby introducing the surface parameters that describe the geometry of the contact surfaces.This method can be applied to study specific problems inherent in railway dynamics suchas the two points of contact scenario. The methodology to look for the candidates tocontact points is fully independent for the wheel tread and flange. Consequently, thecontact point in the flange does not have to be located in the same plane as the contactpoint in the wheel tread. This is a relevant issue, especially when studying the lead and


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lag contact configurations that occur on switch transitions or when dealing with high

angles of attack. This formulation also allows for investigations related with huntinginstability and prediction of wheel climbing, which are very important to study thederailment phenomena. Furthermore, due to the application of an efficientparameterisation procedure, this new formulation can be used for any kind of rail orwheel profiles, new or worn, and still be efficient from the computational point of view,without requiring the use of lookup tables. However, if the concave region of the wheelprofile is expected to play an important role on the vehicle performance, the methodologyproposed must be modified to allow detection of the points of contact with concaveregions and to evaluate the contact forces for conformal contact surfaces.

A comparison of the dynamic analysis results with the results available fromexperimental data and obtained with the commercial program ADAMS/Rail, allowsconcluding that the models and methodologies developed here are not only qualitatively

but also quantitatively correct. It is believed that the differences that are observed resultfrom the uncertainty associated to the experimental tests and from the use of differentprocedures in both computer codes. It is also shown that the formulation integratesefficiently, the main requirements associated with the dynamic analysis of railwayvehicles.

Acknowledgements

The support of Fundao para a Cincia e Tecnologia (FCT) through the Post-Doc grantwith the reference BPD/19066/2004 made this work possible and it is gratefullyacknowledged. The authors want to gratefully acknowledge valuable discussions withProfessor Jos Escalona, University of Seville, and with Professor Ahmed Shabana,University of Illinois at Chicago.

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